The discovery of semi-solid alloys and possible new forming processes can be traced back to the work of Flemings, Mahrabian and their co-workers in the early 1970's. See, for example, U.S. Pat. No. 3,902,544, issued to Flemings, et.al. in 1975. It was found that when molten metals were vigorously agitated during the early stage of solidification, the normally occurring dendritic grain structure was broken into a system of granular solid particles suspended in the remaining liquid matrix.
Since then, extensive studies have been carried out toward developing new forming processes for various metals. In particular, one of the major applications of semi-solid alloys has generally centered on the die casting process. Owing to its high and controllable viscosity, the semi-solid alloy can fill the die cavity in a progressive flow pattern, thus reducing the gas porosity typically caused by the turbulent flow of a superheated molten metal. Such viscous flow behavior of the semi-solid alloy not only ensures sound casting, but also makes possible the task of computer simulation which can be used to predict the casting behavior, improve the efficiency of mold design, and decide the appropriate processing conditions such as temperature and injection speed control.
In pressure die casting, the mold temperature is typically much lower than the melt temperature in order to accelerate the cooling and reduce the cycle time. On the other hand, the solid fraction of the Semi-Solid Metal is relatively high (part of latent heat is removed) and the thermal conductivity of a metal is usually high. As a result, premature freezing could occur if the injection and cooling conditions are inappropriate or if the mold is designed improperly. This means that process control is a key to success in this process and the mold design may become more challenging than it is for conventional pressure die casting. Similar to the injection molding of plastics, the efficiency of mold design and the decision-making for the appropriate processing conditions can be significantly improved with the help of numerical simulation techniques.
Furthermore, most metal parts made by pressure die casting are thin, which means that the half thickness of the part is close to or less than 10% of its planar characteristic length. In this case, the liquid velocity and thermal convection in the transverse direction and the viscous diffusion and thermal condition in the planar direction become negligibly small compared with the other components in the governing equations. It should be noted that although turbulence plays an important role in the conventional die-casting process, its effects can be eliminated in rheomolding by controlling the processing parameters and thus the viscosity of Semi-Solid Metal.
The other crucial factor affecting the rheomolding filling process is the rheology of Semi-Solid Metal. It has been found in previous experiments that the Semi-Solid Metal is pseudo-plastic (shear thinning) at steady state and dilatant (shear thickening) under transient conditions. For instance, when the shear rate applied to Sn-15% Pb Semi-Solid Metal changes suddenly, the viscosity remains at the same value instantaneously and then gradually changes for 20 to 30 seconds before the viscosity reaches another steady state. Since most of the filling time in rheomolding is expected to be less than 1 second, it seems viable to assume that the semi-solid slurry is Newtonian in the mold-filling stage.
In plastic injection molding, the polymer melt is highly viscous and has a low density. Therefore, its inertial effect can be neglected in the flow analysis. This simplification, however, does not apply in rheomolding due to the high injection speed and high density of metals. For a Newtonian liquid flowing between two parallel plates, it can be shown analytically that its velocity magnitude may change with respect to time but the velocity profile remains approximately constant, similar to the one under steady-state conditions.
In order to simulate the complete casting procedure, we need a rheological model to describe the history-dependent properties of semi-solid metal under various process conditions and an efficient tool for fluid dynamic and thermal analyses.
A cavity-filling simulation method must satisfy the following conditions:
1. Stability for a wide range of Reynold's number (from 1 to 1,000 or higher); PA1 2. Can handle multi-moving free surfaces including laminar jets; PA1 3. Convenient in use for complex geometry; PA1 4. Efficient and reasonably accurate; PA1 5. Can be easily incorporated with thermal and stress analyses including phase transformation; and PA1 6. Use primary variables such as velocity and pressure. PA1 v.sub.a is the average velocity vector PA1 .tau. is the shear stress PA1 p is the pressure PA1 .rho. is the density PA1 .mu. is the dynamic viscosity PA1 h.sub.e is the effective half-gap thickness PA1 h.sub.o is the original half-gap thickness PA1 t is time
The reason to choose primary variables instead of stream function or vorticity is for simpler boundary conditions on the die wall in complex domains. Although some existing methods satisfy part of the conditions quite well, they may not be so good in other parts. For instance, the well-known marker-and-cell method and the volume-of-fluid method work satisfactorily except for complex geometry, due to inherent limitations in the finite difference approach. Some other moving mesh methods are able to capture the motions of the free surface more accurately than the fixed mesh methods. The Lagrangian-Eulerian approach, however, still has problems whenever interfaces come into each other or the die wall. Since the filling stage is a dynamic process, the position of the moving surface as well as the velocity and pressure fields is unknown. Thus, an efficient iteration procedure which is independent of the geometry is required. As most of the iteration methods for steady problems (e.g. the spine method) depend on knowing how the free surface moves, and since the mesh must be generated to fit the motion, they are not suitable for general transient processes. Based on these reasons, we felt that a new method was required.